$\int_0^\pi \sin^2 x \cos^3 x \, dx = $ . . . . . . .

The integral value $\int_{-2}^{0} [x^3 + 3x^2 + 3x + 3 + (x + 1)\cos(x + 1)] \, dx$ is

The value of $\int\limits_{\frac{1}{2}}^2 \frac{1}{x} \sin \left( x - \frac{1}{x} \right) dx$ is equal to

The value of $I = \int_{-\pi / 2}^{\pi / 2} |\sin x| \, dx$ is

If $I_{1} = \int_{0}^{1} (1 - x^{50})^{100} dx$ and $I_{2} = \int_{0}^{1} (1 - x^{50})^{101} dx$ such that $I_{2} = \alpha I_{1}$,then $\alpha$ is equal to:

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1+\sqrt{\cot x}} d x=$